How to animate a metal backbone?
Ok, this is what you could try, I hope it fits with your workflow, or at least inspires you to find a better way:


put the body in rest pose
apply the vertebras modifier (array+curve)
separate vertebras mesh into separate vertebras objects
then, create a vertex groups (1 vertex each) on the body back skin, one for each vertebra, which will get "attached" to that vertex later.
then I would create a "child of" constraint for each vertebra, with the right vertex group as "target"


eg: if this is your rigged body (here posed)



select the "body" and create (eg) two vertex group, one "skin" vertex each:



and 



then "attach" each vertebra (here is a sphere) to its vertex group using the "child of" constraint (after adding the constraint, you may need to reposition the "vertebra" where it should be, if it goes elsewhere):



then, when you pose the rig, the body follows the rig, and the vertebras follow the body, without deforming. 

here is my example .blend:


Note: parenting is not the same, will not work well, use this kind of constraint.

edit

after understanding better the rig/model pose, I expand to add details of what I intended...

preface:


I reverted the rig to the "rest position"
you created an "array" of 14 vertebras but not using the array modifier, instead you parented one base vertebra to a curve and used the "frame duplication" feature of such setup, setting the "path animation" frames of the curve to 14.





since what I intended to do above needs 14 real vertebras, not duplicated ones, after setting the mesh "duplication" frames end also to 14 (it was 99, default probably), I used the "make duplicates real" comand, and removed the "base vertebra" parent to the curve.




and 



obtaining 14 real vertebras, with no duplication




then I've hidden all vertebras but the first 2, to keep the example simple to visualize (the whole model is really complex)
after that, I went in edit mode of the body mesh, and created two separate vertex groups, 1 vertex each, each to be the "hook" for a single vertebra.





then, I added to each vertebra the constraint "child of", targeted to each vertex group, and placed them where they should be in rest pose




in solid mode



from the side




now, going back to posed mode, each vertebra sticks to and follows its vertex group, without deforming, but you may need to adjust something to make them move as you desire: the model spine moves quite a lot, and that move vertebreas also... maybe using vertex groups of more than 1 vertex blends the vertebra movement better ... I just tried to keep it simple.




here below you find the updated model with the 2 example vertebras seen in the screenshots above...